How do I make my character's walk animation stop when he jumps?
This happens when you don't add any layer for the jump animation.
What's the meaning of layers in animations?
This is something like Photoshop projects, for example: layer1 is not appearing because there is another layer on top of it which is layer 2.
How can I add a layer or delete a layer in animation?
In the action actuator Here :

This is the layer what the animation is. In your case make the animation layer 2.
What about Deleting a layer?
You can't delete a layer. you can delete the actuator. You can just change the layer.